Clenshaw C W (1955) A note on the summation of Chebyshev series Math. Tables Aids Comput.9 118–120
Cox M G (1974) A data-fitting package for the non-specialist user Software for Numerical Mathematics (ed D J Evans) Academic Press
Cox M G and Hayes J G (1973) Curve fitting: a guide and suite of algorithms for the non-specialist user NPL Report NAC26 National Physical Laboratory
Gentleman W M (1969) An error analysis of Goertzel's (Watt's) method for computing Fourier coefficients Comput. J.12 160–165
1: – IntegerInput
On entry: the number of terms in the series (i.e., one greater than the degree of the polynomial).
2: – Real (Kind=nag_wp) arrayInput
On entry: must be set to the value of the th coefficient in the series, for .
3: – Real (Kind=nag_wp)Input
On entry: , the argument at which the polynomial is to be evaluated. It should lie in the range to , but a value just outside this range is permitted (see Section 6) to allow for possible rounding errors committed in the transformation from to discussed in Section 3. Provided the recommended form of the transformation is used, a successful exit is thus assured whenever the value of lies in the range to .
4: – Real (Kind=nag_wp)Output
On exit: the value of the polynomial.
5: – IntegerInput/Output
On entry: ifail must be set to , . If you are unfamiliar with this argument you should refer to Section 4 in the Introduction to the NAG Library FL Interface for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value is recommended. If the output of error messages is undesirable, then the value is recommended. Otherwise, if you are not familiar with this argument, the recommended value is . When the value is used it is essential to test the value of ifail on exit.
On exit: unless the routine detects an error or a warning has been flagged (see Section 6).
6Error Indicators and Warnings
If on entry or , explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
On entry, and .
Constraint: , where is machine precision.
On entry, .
An unexpected error has been triggered by this routine. Please
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.
The rounding errors committed are such that the computed value of the polynomial is exact for a slightly perturbed set of coefficients . The ratio of the sum of the absolute values of the to the sum of the absolute values of the is less than a small multiple of .
8Parallelism and Performance
e02aef is not threaded in any implementation.
The time taken is approximately proportional to .
It is expected that a common use of e02aef will be the evaluation of the polynomial approximations produced by e02adfande02aff.
Evaluate at equally-spaced points in the interval the polynomial of degree with Chebyshev coefficients, , , , , .
The example program is written in a general form that will enable a polynomial of degree in its Chebyshev series form to be evaluated at equally-spaced points in the interval . The program is self-starting in that any number of datasets can be supplied.